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Subject: Mathematics
Semester: 2
Period: 4
Week: 21
WEEK 21
Class: Grade 12
Age: 17 years
Duration: 40 minutes per period (5 periods)
Subject: Mathematics
Topic: Integration II
Focus:
- Integration of Trigonometric Functions
- Integration by Parts
- Integration by Partial Fractions
- Application of Integration – Simpson’s Rule for Finding Area under the Curve
- Integration of Exponential Functions
SPECIFIC OBJECTIVES
By the end of the lesson, students should be able to:
- Integrate basic trigonometric functions (sine, cosine, tangent).
- Apply the substitution method for integration.
- Use the method of integration by parts.
- Apply partial fractions for integrating rational functions.
- Solve problems involving Simpson’s rule to find areas under curves.
- Integrate exponential functions.
INSTRUCTIONAL TECHNIQUES
- Question and answer
- Guided demonstration
- Discussion
- Practice exercises
- Problem-solving in real-life contexts (e.g., capital market issues)
INSTRUCTIONAL MATERIALS
- Whiteboard and markers
- Integration charts
- Standard integral charts
- Graphing calculator (optional)
- Worksheets with integration exercises
- Real-life application scenarios for Simpson's rule and exponential functions
PERIOD 1: Integration of Trigonometric Functions
PRESENTATION:
Step 1 - Introduction
- Introduce the concept of integrating trigonometric functions.
- Show the standard integrals of sine, cosine, and tangent functions.
Example: ∫sin(x)dx = -cos(x) + C, ∫cos(x)dx = sin(x) + C
Step 2 - Solving Examples
- Solve integration examples on the board, such as:
∫sin(x)dx, ∫cos(x)dx, ∫tan(x)dx
Step 3 - Guided Practice
- Guide students through examples where they integrate trigonometric functions.
- Emphasize recognizing the standard forms of trigonometric functions.
Step 4 - Student Practice
- Allow students to solve similar integration problems individually or in pairs.
- Provide immediate feedback.
EVALUATION:
- Integrate ∫sin(x)dx.
- Integrate ∫cos(x)dx.
- Integrate ∫tan(x)dx.
CLASSWORK:
- ∫sin(2x)dx.
- ∫cos(3x)dx.
- ∫tan(x)dx.
- ∫sec²(x)dx.
- ∫csc(x)dx.
ASSIGNMENT:
- Integrate ∫sin(3x)dx.
- Integrate ∫cos(4x)dx.
- Solve ∫tan(2x)dx.
- Solve ∫sec(x)dx.
- Solve ∫cosec²(x)dx.
PERIOD 2: Integration by Substitution
PRESENTATION:
Step 1 - Introduction to Substitution
- Introduce the concept of substitution in integration.
- Demonstrate how to identify a suitable substitution for integration.
- Example: ∫2x * cos(x²)dx, let u = x² → du = 2xdx
Step 2 - Guided Example
- Solve an example using substitution:
∫2x * cos(x²)dx = ∫cos(u)du = sin(u) + C
Step 3 - Guided Practice
- Guide students through multiple problems, encouraging them to choose substitution variables.
Step 4 - Student Practice
- Let students solve similar problems using substitution.
EVALUATION:
- Solve ∫3x² * sin(x³)dx.
- Solve ∫4x * cos(2x²)dx.
CLASSWORK:
- ∫(x * e^(x²))dx.
- ∫(x² * cos(x³))dx.
- ∫(x * ln(x))dx.
- ∫sin(x²)dx.
- ∫2x * e^(x²)dx.
ASSIGNMENT:
- Solve ∫5x * sin(x²)dx.
- Solve ∫3x * e^(x²)dx.
- Solve ∫x * cos(x²)dx.
- Solve ∫(x * ln(x))dx.
- Solve ∫sin(2x²)dx.
PERIOD 3: Integration by Parts
PRESENTATION:
Step 1 - Introduction to Integration by Parts
- Explain the integration by parts formula:
∫u dv = uv - ∫v du - Demonstrate how to choose u and dv from the integrand.
Step 2 - Guided Example
- Solve an example using integration by parts:
∫x * e^x dx, where u = x and dv = e^x dx.
Step 3 - Guided Practice
- Work through multiple examples with students, solving step by step.
Step 4 - Student Practice
- Let students solve integration by parts problems independently.
EVALUATION:
- Solve ∫x * cos(x)dx.
- Solve ∫ln(x)dx.
- Solve ∫x * e^x dx.
CLASSWORK:
- ∫x² * e^x dx.
- ∫x * ln(x)dx.
- ∫x * sin(x)dx.
- ∫x * cos(x)dx.
- ∫x * e^(x²)dx.
ASSIGNMENT:
- Solve ∫x * ln(x)dx.
- Solve ∫x * e^x dx.
- Solve ∫x² * cos(x)dx.
- Solve ∫x * sin(x)dx.
- Solve ∫x * ln(x²)dx.
PERIOD 4: Integration by Partial Fractions
PRESENTATION:
Step 1 - Introduction to Partial Fractions
- Explain the process of decomposing rational functions into partial fractions.
- Show standard techniques for decomposing, such as factoring the denominator.
Step 2 - Guided Example
- Demonstrate the decomposition and integration of rational functions, e.g.,
∫(1 / (x² - 1)) dx.
Step 3 - Guided Practice
- Work through examples together, focusing on factorizing denominators and breaking them into partial fractions.
Step 4 - Student Practice
- Let students practice decomposing and integrating rational functions.
EVALUATION:
- Solve ∫(1 / (x² + 1)) dx.
- Solve ∫(1 / (x² - 4)) dx.
CLASSWORK:
- ∫(1 / (x² - 1)) dx.
- ∫(1 / (x³ - 3x)) dx.
- ∫(1 / (x² + x - 6)) dx.
- ∫(1 / (x² + 1)) dx.
- ∫(1 / (x² - 2x - 3)) dx.
ASSIGNMENT:
- Solve ∫(1 / (x² - 1)) dx.
- Solve ∫(1 / (x² + 2x + 1)) dx.
- Solve ∫(1 / (x³ - x)) dx.
- Solve ∫(1 / (x² + 2x - 3)) dx.
- Solve ∫(1 / (x² + 4x + 4)) dx.
PERIOD 5: Application of Integration – Simpson’s Rule
PRESENTATION:
Step 1 - Introduction to Simpson’s Rule
- Introduce Simpson’s Rule as a method for approximating the area under a curve.
- Formula:
A ≈ (h / 3) [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xn)]
where h is the step size.
Step 2 - Guided Example
- Demonstrate how to apply Simpson’s Rule to find the area under a curve for a given function, e.g.,
A ≈ (h / 3) [f(0) + 4f(1) + 2f(2) + ... + f(4)].
Step 3 - Guided Practice
- Solve an example using Simpson's Rule, guiding students step by step.
Step 4 - Student Practice
- Let students solve problems on Simpson's Rule for area under curves.
EVALUATION:
- Apply Simpson's Rule to approximate the area under the curve y = x² between x = 0 and x = 2.
- Apply Simpson's Rule to find the area under y = cos(x) from x = 0 to x = π.
CLASSWORK:
- Use Simpson's Rule to find the area under y = x³ from x = 0 to x = 1.
- Apply Simpson's Rule for y = sin(x) from x = 0 to x = π.
ASSIGNMENT:
- Approximate the area under the curve y = e^x from x = 0 to x = 1 using Simpson's Rule.
- Use Simpson's Rule to find the area under y = x² + 3 from x = 1 to x = 3.